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Supergravity in Two Spacetime Dimensions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server DISSERTATION Supergravity in Two Spacetime Dimensions AThesis Presented to the Faculty of Science and Informatics Vienna University of Technology Under the Supervision of Prof. Wolfgang Kummer Institute for Theoretical Physics In Partial Fulfillment Of the Requirements for the Degree Doctor of Technical Sciences By Dipl.-Ing. Martin Franz Ertl Streitwiesen 13, 3653 Weiten, Austria E-mail: [email protected] Vienna, January 31, 2001 Acknowledgements I am most grateful to my parents for rendering my studies possible and their invaluable support during this time. Further thanks go to my sister Eva and my brother in law Fritz, as well as to my grandmother and my aunts Maria and Gertrud. Let me express my gratitude to Prof. Wolfgang Kummer who was the supervisor of both my diploma thesis and my PhD thesis, and whose FWF projects established the financial background for them. He advised and assisted me in all problems encountered during this work. I gratefully acknowledge the profit I gained from collaborations with Michael Katanaev (Steklov Institute Moskau) und Thomas Strobl (currently at Friedrich-Schiller University Jena). Furthermore, I also thank Manfred Schweda and Maximilian Kreuzer for financial support. Finally, I am also very grateful to all the members of the institute for constantly encouraging me and for the nice atmosphere. Special thanks go to Herbert Balasin, Daniel Grumiller, Dragica Kahlina, Alexander Kling, Thomas Kl¨osch, Herbert Liebl, Mahmoud Nikbakht-Tehrani, Anton Reb- han, Axel Schwarz and Dominik Schwarz. This work was supported by projects P 10.221-PHY and P 12.815-TPH and in its final stage by project P 13.126-PHY of the FWF (Osterreichischer¨ Fonds zur F¨orderung der wissenschaftlichen Forschung). Abstract Supersymmetry is an essential component of all modern theories such as strings, branes and supergravity, because it is considered as an indispensable ingredient in the search for a unified field theory. Lower dimensional models are of particular importance for the investigation of physical phenomena in a somewhat simplified context. Therefore, two-dimensional supergravity is discussed in the present work. The constraints of the superfield method are adapted to allow for su- pergravity with bosonic torsion. As the analysis of the Bianchi identities reveals, a new vector superfield is encountered besides the well-known scalar one. The constraints are solved both with superfields using a special decom- position of the supervielbein, and explicitly in terms of component fields in a Wess-Zumino gauge. The graded Poisson Sigma Model (gPSM) is the alternative method used to construct supersymmetric gravity theories. In this context the graded Jacobi identity is solved algebraically for general cases. Some of the Poisson algebras obtained are singular, or several potentials contained in them are restricted. This is discussed for a selection of representative algebras. It is found, that the gPSM is far more flexible and it shows the inherent ambiguity of the supersymmetric extension more clearly than the superfield method. Among the various models spherically reduced Einstein gravity and gravity with torsion are treated. Also the Legendre transformation to eliminate auxiliary fields, superdilaton theories and the explicit solution of the gPSM equations of motion for a typical model are presented. Furthermore, the PSM field equations are analyzed in detail, leading to the so called ”symplectic extension”. Thereby, the Poisson tensor is extended to become regular by adding new coordinates to the target space. For gravity models this is achieved with one additional coordinate. Finally, the relation of the gPSM to the superfield method is established by extending the base manifold to become a supermanifold. Contents 1 Introduction 1 2 Supergravity with Superfields 10 2.1Supergeometry.......................... 10 2.1.1 Superfields........................ 11 2.1.2 Superspace Metric . ................... 13 2.1.3 Linear Superconnection . ............ 13 2.1.4 Einstein-CartanVariables................ 14 2.1.5 SupercurvatureandSupertorsion............ 16 2.1.6 BianchiIdentities..................... 17 2.2DecompositionoftheSupervielbein............... 18 2.3SupergravityModelofHowe................... 19 2.3.1 ComponentFields.................... 19 2.3.2 SymmetryTransformations............... 22 2.3.3 SupertorsionandSupercurvature............ 23 2.4LorentzCovariantSupersymmetry............... 24 2.5NewSupergravity......................... 27 2.5.1 Bianchi Identities for New Constraints . 27 2.5.2 SolutionoftheConstraints............... 30 2.5.3 PhysicalFieldsandGaugeFixing............ 31 2.5.4 Component Fields of New Supergravity . 32 2.5.5 SupertorsionandSupercurvature............ 37 3 PSM Gravity and its Symplectic Extension 38 3.1PSMGravity........................... 38 3.2ConservedQuantity....................... 39 3.3SolvingtheEquationsofMotion................ 40 3.4 Symplectic Extension of the PSM . ............ 42 3.4.1 SimpleCase........................ 43 3.4.2 GenericCase....................... 44 3.4.3 UniquenessoftheExtension............... 46 3.5 Symplectic Geometry . ................... 47 3.6 Symplectic Gravity . ................... 48 iii Contents iv 3.6.1 Symmetry Adapted Coordinate Systems . 50 4 Supergravity from Poisson Superalgebras 52 4.1GradedPoissonSigmaModel.................. 53 4.1.1 OutlineoftheApproach................. 53 4.1.2 DetailsofthegPSM................... 55 4.2SolutionoftheJacobiIdentities................. 58 4.2.1 Lorentz-Covariant Ansatz for the Poisson-Tensor . 58 4.2.2 RemainingJacobiIdentities............... 60 4.3Targetspacediffeomorphisms.................. 66 4.4ParticularPoissonSuperalgebras................ 68 4.4.1 Block Diagonal Algebra . ............ 68 4.4.2 NondegenerateChiralAlgebra............. 70 4.4.3 DeformedRigidSupersymmetry............ 70 4.4.4 DilatonPrepotentialAlgebra.............. 71 4.4.5 Bosonic Potential Linear in Y .............. 74 4.4.6 GeneralPrepotentialAlgebra.............. 78 4.4.7 Algebra with u(φ, Y )and˜u(φ, Y )............ 79 4.5SupergravityActions....................... 81 4.5.1 FirstOrderFormulation................. 81 4.5.2 Elimination of the Auxiliary Fields XI ......... 82 4.5.3 SuperdilatonTheory................... 84 4.6ActionsforParticularModels.................. 86 4.6.1 Block Diagonal Supergravity . ............ 86 4.6.2 ParityViolatingSupergravity.............. 88 4.6.3 DeformedRigidSupersymmetry............ 88 4.6.4 DilatonPrepotentialSupergravities........... 89 4.6.5 Supergravities with Quadratic Bosonic Torsion . 92 4.6.6 N =(1, 0)DilatonSupergravity............. 95 4.7SolutionoftheDilatonSupergravityModel.......... 96 5 PSM with Superfields 101 5.1OutlineoftheApproach.....................101 5.1.1 BaseSupermanifold...................101 5.1.2 Superdiffeomorphism...................103 5.1.3 ComponentFields....................103 5.2RigidSupersymmetry......................104 5.3SupergravityModelofHowe...................107 6 Conclusion 110 A Forms and Gravity 115 Contents v B Conventions of Spinor-Space and Spinors 119 B.1Spin-ComponentsofLorentzTensors..............122 B.2DecompositionsofSpin-Tensors.................124 B.3PropertiesoftheΓ-Matrices...................127 Chapter 1 Introduction The study of diffeomorphism invariant theories in 1 + 1 dimensions has for quite some time been a fertile ground for acquiring some insight into the unsolved problems of quantum gravity in higher dimensions. Indeed, the whole field of spherically symmetric gravity belongs to this class, from d-dimensional Einstein theory to extended theories like the Jordan-Brans- Dicke theory [? , ? , ? , ? , ? ] or ‘quintessence’ [? , ? , ? , ? , ? , ? ] which may now seem to obtain observational support [? , ? , ? , ? , ? , ? ]. Also, equivalent formulations for 4d Einstein theory with nonvanishing torsion (‘teleparallelism’ [? , ? , ? , ? , ? , ? , ? , ? , ? , ? , ? , ? ]) and alternative theories including curvature and torsion [? ] are receiving increasing attention. On the other hand, supersymmetric extensions of gravity [? , ? , ? , ? , ? ] are believed to be a necessary ingredient for a consistent solution of the problem of quantizing gravity, especially within the framework of string/brane theory [? , ? , ? ]. These extensions so far are based upon bosonic theories with vanishing torsion. Despite the fact that so far no tangible direct evidence for supersymme- try has been discovered in nature, supersymmetry [? , ? , ? , ? ] managed to retain continual interest within the aim to arrive at a fundamental ‘theory of everything’ ever since its discovery: first in supergravity [? , ? , ? , ? , ? ]ind = 4, then in generalizations to higher dimensions of higher N [? ], and finally incorporated as a low energy limit of superstrings [? ]orofeven more fundamental theories [? ] in 11 dimensions. Even before the advent of strings and superstrings the importance of studies in 1 + 1 ‘spacetime’ had been emphasized [? ] in connection with the study of possible superspace formulations [? ]. To the best of our knowledge, however, to this day no attempt has been made to generalize the supergravity formulation of (trivial) Einstein-gravity in d = 2 to the consideration of two-dimensional (1, 1) supermanifolds for which the condi- tion of vanishing (bosonic) torsion is removed. Only attempts to formulate 1 Chapter 1. Introduction 2 theories with higher powers of curvature
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